Place an oven rack in the center of the oven and preheat to 350 degrees F.
For the chips: Brush the breads with the oil using a pastry brush and sprinkle with the Parmesan. Cut each bread into 32 triangles and arrange in a single layer, cheese-side-up, on 2 baking sheets. Bake until light golden, 10 to 12 minutes.
For the dip: Bring the tomato-basil sauce to a simmer over medium heat in a medium saucepan. Cook for 5 minutes. Remove the saucepan from the heat and whisk in the gorgonzola and mascarpone until incorporated. Pour the dip into a serving bowl and serve the chips alongside.